Odds Ratio vs. Relative Risk: Choosing the Right Measure

1. Observational Study Designs: The Deciding Factor

To understand which measure to use, we first need to distinguish between the two primary types of observational studies you'll encounter. These designs differ fundamentally in how subjects are sampled.

To summarize the key points from the video:

• Cohort Study: You start with a population, identify an exposed group (e.g., smokers) and a non-exposed group (e.g., non-smokers), and follow them forward in time to see who develops the outcome (e.g., lung cancer). You are sampling based on exposure.
• Case-Control Study: You start with the outcome. You identify a group of individuals who already have the disease (cases) and a separate group who do not have the disease (controls). You then look backward in time (retrospectively) to determine their past exposure history. You are sampling based on the outcome.

This distinction in sampling strategy is the fundamental reason why we can't always use the same measures of association.

2. Matching the Measure to the Design

Now, let's connect our measures—Relative Risk and Odds Ratio—to these study designs.

For Cohort Studies: You Can Choose

In a cohort study, because you follow a disease-free population over time, you can directly measure the number of new cases that appear in each group. This allows you to calculate the incidence (i.e., the risk) of the disease.

• Risk in the exposed group = $a/(a+b)$
• Risk in the unexposed group = $c/(c+d)$

Since you can calculate risk, you can calculate the Relative Risk (RR), which is simply the ratio of these two risks.

In a cohort study, you can also calculate the odds ratio. So, which should you report?

• Relative Risk is generally preferred for cohort studies. Its interpretation is more direct and intuitive: "The risk of developing the disease is X times higher in the exposed group."
• Odds Ratio is still a valid measure. While its interpretation is less direct ("The odds of developing the disease..."), it becomes crucial for the statistical modeling techniques (like logistic regression) that we'll cover later.

Key Takeaway: In a cohort study, you have a choice. For straightforward reporting of association, RR is often best.

For Case-Control Studies: You Have No Choice

Now, consider the case-control design. Here, you, the researcher, decide how many cases and controls to include. For instance, you might recruit 100 people with lung cancer and 100 people without it. The 50% disease rate in your sample is artificial and does not reflect the true risk of lung cancer in the general population.

Because you cannot calculate the true incidence from the sample, you cannot calculate the Relative Risk. This is the critical failure point for RR.

This is where the OR's invariance property becomes essential. Even though we can't calculate the risk of disease, we can calculate the odds of exposure among the cases and compare them to the odds of exposure among the controls. As we learned last lesson, the resulting odds ratio is numerically identical to the odds ratio of disease we would have gotten in a cohort study.

Key Takeaway: In a case-control study, the Odds Ratio is the only valid measure of association you can compute directly.

3. The OR as an Estimate of the RR: The Rare Disease Assumption

We know we must use the OR in a case-control study, but we are often fundamentally interested in risk. So, when can the OR from a case-control study be used as a stand-in for the RR we wish we could calculate?

This is possible under the rare disease assumption.

As we touched on in the last lesson, when an outcome is rare, the risk ($\pi$) and the odds ($\pi /(1-\pi )$) are very close in value. Therefore, for rare diseases, the Odds Ratio provides a good approximation of the Relative Risk.

However, if the disease is common, the OR will always overestimate the RR (i.e., the OR will be further from the null value of 1.0 than the RR).

This is a critical point for interpretation: if you calculate an OR of 4.0 for a common disease, you cannot claim the risk is four times higher. The odds are four times higher, but the increase in risk is less.

4. An Advanced Note: When the Rare Disease Assumption Isn't Needed

For your functional understanding as a data scientist, it's useful to know that the "rare disease assumption" is a bit of a simplification. More advanced sampling methods in case-control studies can bypass this requirement.

The key insight is that if controls are sampled concurrently with when cases are identified (incidence density sampling), the OR directly estimates the rate ratio (a type of relative risk) without needing to assume the disease is rare. This is a more robust design, and knowing it exists deepens your understanding of how these studies can be engineered to yield powerful results.

5. Summary and Review

The choice between RR and OR is driven by the study design. Let's consolidate this with a summary.

To solidify these concepts, a final reading will provide a comprehensive summary.

Conclusion

You have now completed the first module on the foundations of categorical analysis. You can distinguish between different data types, build contingency tables, and—most importantly—calculate and correctly apply the three main measures of association: the difference in proportions, relative risk, and odds ratio.

Key Takeaways from this lesson:

• The choice between RR and OR is primarily dictated by the study design.
• In cohort studies, you can calculate both, but RR is preferred for its direct interpretability.
• In case-control studies, you cannot calculate incidence, so OR is the only valid measure of association.
• The OR from a case-control study can approximate the RR if the disease is rare. If the disease is common, the OR will overestimate the RR.

Preview of the Next Lesson:

So far, we have described and measured associations. But how do we know if an observed association is statistically significant or if it could have just occurred by chance? In the next module, we will move from descriptive measures to inferential statistics, starting with the foundational Pearson Chi-Square Test of Independence.